Materials and Design 31 (2010) 296–305
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Materials and Design
journal homepage: www.elsevier.com/locate/matdes
Influence of continuum damage mechanics on the Bree’s diagram
of a closed end tube
A. Nayebi *
Department of Mechanical Engineering, Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history:
Received 30 April 2009
Accepted 11 June 2009
Available online 13 June 2009
Keywords:
Continuum damage mechanics
Nonlinear kinematic hardening
Bree’s diagram
Shakedown
Ratcheting
Return mapping algorithm
a b s t r a c t
This paper extends the Bree’s cylinder behaviors, which is subjected to the constant internal pressure and
cyclic temperature gradient loadings, with considering continuum damage mechanics coupled with nonlinear kinematic hardening model. The Bree’s biaxial stress model is modified using the unified damage
and the Armstrong–Frederick nonlinear kinematic hardening models. With the help of the return mapping algorithm, the incremental plastic strain in axial and tangential directions is obtained. Continuum
damage mechanics approach can be used to extend the Bree’s diagram to the damaging structures and
reduce the plastic shakedown domain. Kinematic hardening behavior was considered in the material
model which shifts the ratcheting zone. The role of the material constants in the Bree’s diagram is also
discussed.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
The need for a suitable constitutive model to predict the shakedown or cyclic failure (ratcheting) of structures under cyclic loading is increasing in many industries, and in order to better obtain
the structures behaviors, many researchers tried to develop improved constitutive models [1]. Although, the proposed models
simulate well the uniaxial ratcheting responses, there are other
factors that influence the biaxial stress cyclic loading. Shakedown
loads and different behaviors of structures were also studied by
many authors with the plasticity and cyclic plasticity models [2].
In particular, the two-bar problem and the Bree’s cylinder were
studied under different loading conditions and materials behaviors. Parkes [3] studied thermal ratcheting in an aircraft wing
resulting from the cyclic thermal stresses superimposed on the
normal wing loads. Miller [4] showed that the material strain hardening reduces considerably the strains due to ratcheting in the
two-bar structure. Jiang and Leckie [5] presented a method for direct determination of the steady solutions in shakedown analysis
with application to the two-bar problem. Bree [6] analyzed the
elastic–plastic behavior of a thin cylindrical tube subjected to constant internal pressure and cyclic temperature gradient across the
tube thickness. A simple one-dimensional model, a linear temperature drop distribution across the cylinder thickness and an elastic–perfectly-plastic material model were assumed in his
analysis. Later, he used a biaxial stress model and obtained a more
* Tel.: +98 711 6133029.
E-mail address: [email protected]
0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.matdes.2009.06.014
complete interaction diagram (Bree’s diagram) for a closed tube
[7]. In that study the material was assumed to be perfect plastic.
The Poisson’s effect was also neglected. The interaction diagram
proposed by Bree received adequate investigation by a number of
researchers [8,9]. His one-dimensional diagram is a part of ASME
boiler and pressure vessel code [8] (Fig. 1). Mulcahy [9] improved
the Bree’s analysis with incorporating a linear kinematic hardening
model in the analysis of a beam element.
Classical shakedown theory of Melan–Koiter for elastic–
perfectly-plastic bodies has been well established in the literature
[10]. Melan procedure for the shakedown theorem can also be extended to encompass the linear kinematic hardening, but one
encounters mathematical difficulty in treating more general cases,
and the procedure does not apply straightforwardly for the case of
Prager’s linear kinematic hardening. It seems that some additional
assumptions as well as further mathematical tricks are needed to
deal with the general kinematic hardening. Recently, Abdalla
et al. [11] proposed a simple shakedown method with perfect plasticity material behavior to study the Bree’s cylinder problem and
the 90° pipes bending.
Very limited work has been done on damage related to the
interaction diagram. The shakedown theory has been extended to
include hardening and damage by Hachemi and Weichert [12]
and Druyanov and Roman [13]. However, the extensions should
be made often at the expense of losing certain fine features of
the classical plasticity theory and shakedown theorems. Without
the theorems in Melan–Koiter sense, which are valid only with
certain restrictions, generally in practice one has to implement
numerical incremental analysis to check for shakedown of a
A. Nayebi / Materials and Design 31 (2010) 296–305
297
Nomenclature
A
~
A
C
Dc
E
~
E
ez
eh
f
FK1
K2
nij
P
p_
Dp
q
Q
r
Rt
Sz
Sh
t
T
DT
virgin surface
resistant effective surface
NLKH model’s constant
interatomic decohesion damage parameter
elastic modulus
effective elastic modulus
nondimensional axial strain
nondimensional tangential strain
yield function
dissipative potential function
mean tangential strain
mean axial strain
outward normal to the yield surface
internal pressure
equivalent plastic strain rate
equivalent plastic strain increment
material damage constant
material damage constant
mean radius
triaxiality function
nondimensional axial stress
nondimensional tangential stress
thin wall thickness
temperature
temperature difference between the inner and outer
surfaces
DTx
x
X
X’
Y
temperature difference in each radius
coordinate
back stress tensor
deviatoric back stress tensor
associate thermodynamics damage variable tensor
expansion coefficient
NLKH model’s constant
incremental plastic multiplier
nondimensional thickness
plastic strain tensor
thermal strain tensor
plastic damage threshold strain
stress tensor
effective stress tensor
deviatoric stress tensor
equivalent von-Mises stress
hydrostatic stress
nondimensional mechanical stress parameter
yield stress
nondimensional temperature difference
nondimensional temperature difference increment
Poisson’s ratio
damage parameter
a
c
dk
g
ep
eT
epD
r
~
r
r0
req
rH
rP
ry
s
Ds
t
-
structure under specific loading histories. Recently, Nayebi and El
Abdi [14] and Kang et al. [15] used continuum damage mechanics
(CDM) to predict the material behavior, including ratcheting and
shakedown in 1-D analysis.
In this research, the effect of continuum damage mechanics on
the Bree’s diagram of a thin cylinder structure under specific loading histories is studied. Nonlinear kinematic hardening (NLKH)
theory is coupled with continuum damage mechanics in order to
model the behavior of the Bree’s cylinder. A unified damage
mechanics model, which is also appropriate for low cyclic loading,
is used. During each loading, the damage analysis is performed. An
iterative method is used to analyze the cylinder under the cyclic
thermal and constant mechanical loads. The model extends Bree’s
2-D diagram to incorporate damage effects. The proposed method
can also be applied to other structures subjected to cyclic thermal
and constant mechanical loadings. The results illustrate the influence of material damage on the behaviors of structures under cyclic loading in comparison with the confirmed results on
undamaged structures.
material can then be represented by the constitutive equations of
the virgin material where the usual stress tensor, r, is replaced by
~ defined by
the effective stress r
2. Constitutive behavior relations
where k_ is calculated from the constitutive equations of plasticity
coupled with the damage deduced from the dissipative potential
_ , and
function, F-. Y is the associate variable of the damage rate, epD is the plastic damage threshold strain. Also, many experimental
results indicated that Fx must be a nonlinear function of Y [16]
2.1. Continuum damage mechanics
According to the applied theory of damage mechanics, microscopic change in a material element of surface A develops into
macroscopic defect as a result of loading. In the damaged state,
~ (Fig. 2), from which the isotropic
the new area is denoted by A
damage variable - is defined as [16]
~
AA
-¼
;
A
ð1Þ
where - may be considered as an internal state variable characterizing the irreversible deterioration of a material in the thermodynamic sense. Following this theory, the behavior of a damaged
~¼
r
r
;
1-
ð2Þ
where the value - = 0 corresponds to the undamaged state, 2 (0, Dc) corresponds to a partly damaged state, and - = Dc defines
the element state rupture by interatomic decohesion (Dc 2 [0, 1]). In
the sequel, superposed tilde indicates quantities related to the damaged state of the material.
From a physical point of view, the material degradation involves
the initiation, growth and coalescence of micro-cracks or microvoids generally induced by large plastic strains. This phenomenon
is called ductile plastic damage and leads to plastic (ductile) fracture. Many observations and experiments indicated that the damage is also governed by the plastic strain which is introduced into
_ as [16]
the model through the plastic multiplier k,
-_ ¼ k_
F- ¼
@F @Y
if
ep epD ;
qþ1
Q
Y
ðq þ 1Þð1 -Þ Q
ð3Þ
ð4Þ
from which Eq. (3) reduces to
-_ ¼
q
Y
p_
Q
ð5Þ
where Q and q are material parameters and p_ is the equivalent plastic strain rate. According to Lemaitre and Desmorat [16]
298
A. Nayebi / Materials and Design 31 (2010) 296–305
Fig. 1. Bree’s diagram for a 1-D tube model [6].
8
~2
>
< Y ¼ req2ERt
where p_ is the accumulated plastic strain rate given by
2 ;
>
: Rt ¼ 2 ð1 þ tÞ þ 3ð1 2tÞ rH
3
req
ð6Þ
p_ ¼
0
0 1/2
where Rt is the triaxiality function of stress, req = (3/2r :r ) is the
von-Mises equivalent stress (r0ij ¼ rij rH dij and: indicates the inner
product of two tensors), rH = 1/3tr(r) denotes the hydrostatic stress
(dij is the Kronecker unit tensor), E is Young modulus, and t is the
Poisson’s ratio. Using Eqs. (2) and (6), one can reduce the damage
law (Eq. (5)) to the unified damage law for low cycle fatigue as [16]
-_ ¼
r
2
eq Rt
2EQð1 -Þ2
!q
1
2 p p 2
e_ : e_
3
So, the damage parameter is related to the equivalent plastic strain
and is coupled with the plasticity. Assuming the von-Mises yield
criteria, the yield surface can be rewritten by replacing the stress
~, as
with the effective stress r
f ð~
r; XÞ ¼ f
_
p;
ð7Þ
ð8Þ
r
;X ;
1-
where X is the back stress.
ð9Þ
299
A. Nayebi / Materials and Design 31 (2010) 296–305
where C and c are material parameters and
Surface without
voids and cracks
dk ¼ ð1 -Þdp
ð15Þ
3. Closed-tube biaxial-stress model based on CDM
A
Fig. 2. Definition of the surface (A) and the effective resistant surface (Ã).
The studied pressure vessel was assumed to be a thin cylinder
with the mean radius r, and the wall thickness, t. It is closed at both
ends. The thin cylinder is subjected to an internal pressure P and a
heat flux through its internal surface (Fig. 3). The temperature difference decreases linearly across the wall thickness and is changed
cyclically between DT and zero. It is assumed that the cylindrical
shell is very long and the end effects and the curvature can be
neglected.
The axial and hoop strains, ez and eh, are spatially constant [8].
This is because of the bending prevention; however they change
during each cycle. The axial and hoop stresses, rz and rh, vary
across the thickness and are only dependent on the coordinate x
shown in the Fig. 3a. Using the equilibrium conditions, it is required that
Z
t
2
2.2. Nonlinear kinematic hardening
Nonlinear kinematic hardening is introduced using the differential form of the governing equations for the kinematic variables.
Based on the von-Mises yield criteria, the equation of the yield surface is written as
f ¼ J 2 ð~
r XÞ ry ¼
12
3 0
r0 X 0 Þ ry ¼ 0;
ð~
r X 0 Þ : ð~
2
ð10Þ
where X is the back stress defining the position of the yield surface
and ry characterizes the size of the surface. The plastic flow follows
the normality rule
dep ¼
dk @f
3 dk
ð~
r0 X 0Þ
:
¼
1 - @r 2 1 - ð~
r0 X 0 Þeq
ð11Þ
The plastic multiplier dk is derived from the consistency condition,
f = df = 0, if plastic flow occurs. Different kinematic hardening models are available for the plastic analysis of structures. Sehitoglu et al.
[17] pointed out that the material model was critical for the stress
analysis of a damaged component. Many cyclic plasticity models
were tested under different cyclic loadings [18–21]. A number of
loading responses may be predicted by these models, while they fail
to predict other types of cyclic loading conditions. Accordingly, it is
difficult to understand which case of loading should be simulated
with which model. In this paper, the Armstrong–Frederick nonlinear kinematic hardening model [22] coupled with continuum damage mechanics is used to determine the structure behavior under
cyclic loadings.
If the plastic strain ðepij Þ and the back stress tensor (Xij) are assumed as the internal variables, the evolution equations are
t
2
t
2
Z
t
2
rh dx ¼ Pr;
rz dx ¼
Pr
2
x
DT x ¼ DT;
t
ð18Þ
where DT is the temperature difference between inner and outer
surface of the tube. The mean temperature in each cycle was assumed sufficiently low so that creep effects could be ignored.
Fig. 3 demonstrates the loading steps.
R
ΔTout
x
t
ΔTin
(a)
Thermal loading
Mechanical loading
depij ¼ dknij ;
r~ 0ij X 0ij
3 1
~ ij X ij k
2 1- kr
where nij ¼
¼
is the outward normal to the
yield surface and dk is the incremental plastic multiplier calculated from the consistency condition
@f
@f
@f
drij þ
dX ij þ
d- ¼ 0;
@ rij
@X ij
@-
ð13Þ
(b)
(b) Nonlinear kinematic hardening model
2
Cð1 -Þdepij þ cX ij dk;
3
Loading
ð12Þ
@f
@ rij
dX ij ¼
ð17Þ
Consequently, every element of the tube is subjected to the mean
in axial direction and Prt in hoop direction. The temperature
stress Pr
2t
difference in start up half cycle is assumed to vary linearly with respect to x. It is zero in the thin cylinder mid-wall and at the shutdown second half cycle
(a) Flow rule:
df ¼
ð16Þ
ð14Þ
Time
Fig. 3. (a) Tube geometry and applied temperature gradient across the tube wall
thickness, t, (b) constant mechanical and cyclic temperature gradient loadings
history.
300
A. Nayebi / Materials and Design 31 (2010) 296–305
Since bending is prevented in both hoop and axial direction, eh
and ez are constants across the tube thickness. Therefore, the total
strain in both directions and for every loading is constant
Table 1
Materials models constants [16].
E
t
ry
c
C
Q
q
epD
Dc
eh ¼ K 1
ez ¼ K 2
134 GPa
0.3
85 MPa
250
5500 MPa
0.6 MPa
2
0.2 ey
0.2
ð19Þ
ð20Þ
Using the strain partition principle, the strains in hoop and axial
direction have three parts (in small strain hypothesis): elastic, thermal and plastic strains. The total strains in two directions are:
e
h
p
h
T
h
r~ h
eh ¼ e þ e þ e ¼
ez ¼ eez þ eTz þ epz
t
r~ z
p
h
þ aDT x þ e
E
E
r~ z
r~ h
þ aDT x þ epz
¼
t
E
E
ð21Þ
ð22Þ
where ee, eT and ep are elastic, thermal and plastic strains, respec~ z are effective tangential and longitudinal stresses
~ h and r
tively. r
according to the continuum damage mechanics. They are defined
in the Section 2.1. Eqs. (18)–(22) were solved to obtain hoop and axial stresses as
E
r~ h ¼
ðK 1 þ tK 2 ð1 þ tÞaDT x eph tepz Þ;
1 t2
E
r~ z ¼
ðK 2 þ tK 1 ð1 þ tÞaDT x epz teph Þ
1 t2
ð23Þ
where
(
rp ¼ tPrry
DT x
s ¼ eyað1
tÞ
ð32Þ
Plastic strains can be determined from the constitutive relations
that include the yield criterion, the normality rule, and the back
stress model. The yield function for the biaxial stress using vonMises criterion is
~ h X h Þ2 þ ðr
~ z X z Þ2 ðr
~ h X h Þðr
~ z X z Þ r2y ¼ 0
f ¼ ðr
ð33Þ
By the normality rule, Eq. (11), the relation between hoop and axial
plastic strain increments can be obtained as
ð24Þ
Substituting Eqs. (19) and (20) into Eqs. (21) and (22) and using Eqs.
(16) and (17), we can determine the constants K1 and K2 as
Rt
1 2t PrE þ t2 ð1 -ÞðaDT x þ eph Þdx
2
K1 ¼
;
R 2t
t ð1 -Þdx
2
1
Rt
t PrE þ t2 ð1 -ÞðaDT x þ epz Þdx
2
2
K2 ¼
:
R 2t
t ð1 -Þdx
ð25Þ
ð26Þ
2
In this part the following dimensionless parameters are presented:
8
ep
ep
>
eph ¼ ehy ; epz ¼ ezy ; g ¼ xt ;
>
>
<
Sh ¼ rryh ; Sz ¼ rryz ; rp ¼ tEPrey
>
>
>
: k ¼ K 1 ; k ¼ K 2 ; e ¼ ry
1
2
y
ey
ey
E
ð27Þ
Using these dimensionless parameters, we can simplify Eqs. (25)
and (26).
k1 ¼
1 2t
rp þ
R 12
1
2
ð1 -Þ aDT x =ey þ eph dg
R 12
ð1 -Þdg
1
R 12
t rp þ 1 ð1 -Þ aDT x =ey þ epz dg
2
2
k2 ¼
R 12
1 ð1 -Þdg
ð28Þ
1
2
ð29Þ
2
Substituting the above relations for k1 and k2 into Eqs. (23) and (24),
we can obtain the nondimensional stresses as a function of plastic
strains as
0
r þ 1
B p ð1t2 Þ
Sh ¼ ð1 -Þ@
hR 1
i
ð1 -Þðeph þ tepz þ sÞdg
R 12
1 ð1 -Þdg
2
2
1
2
1
ðep þ tepz Þ s
ð1 t2 Þ h
0
hR 1
i
p
p
1
1
2
r
p þ ð1t2 Þ
1 ð1 -Þðez þ teh þ sÞdg
2
B
2
Sz ¼ ð1 -Þ@
R 12
1 ð1 -Þdg
2
1
ðep þ teph Þ s
ð1 t2 Þ z
ð30Þ
ð31Þ
Fig. 4. Elastic shakedown behavior of the thin cylinder for smax = 1.976 and rp = 0.5,
material models constants are given in Table 1. (a) variation of the maximum
nondimensional tangential plastic strain, eph , at the outer surface of the thin cylinder
as a function of the number of thermal loading cycles and (b) normalized equivalent
stress versus normalized equivalent plastic strain at the outer surface.
A. Nayebi / Materials and Design 31 (2010) 296–305
~z r
~ h 2X z þ X h
depz 2r
¼
~h r
~ z 2X h þ X z
deph 2r
ð34Þ
In order to obtain the variations of the plastic strain and stresses,
Eqs. (7), (14), (30), (31), (33), and (34) are to be solved.
4. Numerical procedure
In order to solve Eqs. (30), (31), (33), and (34), the return mapping algorithm RMA [23,24] was used. This method represents a
well established integration scheme to integrate the rate constitutive equations. This method consists of an elastic trial and plastic
corrector step. When the yield function is convex, i.e. fntrial > fn at
time step n, the elastic trial step is employed to characterize the
plastic loading/unloading state of the material using the algorithmic Kuhn–Tucker conditions
fn 0;
Dkn 0;
Dkn f n ¼ 0:
ð35Þ
where Dkn is the increment of the plastic multiplier.
Fig. 5. Plastic shakedown behavior of the thin cylinder for smax = 2.196 and rp = 0.5,
material models constants and thin cylinder geometry are given in Table 1. (a)
Variation of the maximum nondimensional tangential plastic strain, eph at the outer
surface of the thin cylinder as a function of the number of thermal loading cycles
and (b) normalized equivalent stress versus normalized equivalent plastic strain at
the outer surface.
301
At each time step, the yield function is evaluated at the trial
elastic step, in order to determine whether the yield occurs or
not. If the trial yield function is less than zero, then the material
is assumed to be elastic or plastic but under elastic unloading.
Otherwise, the material is subjected to the plastic loading. The
normality rule is used to define the plastic strain increment in
the radial RMA. The yield condition is used to determine the
incremental value of the plastic multiplier Dcn at the current
time step n. Since a nonlinear kinematic hardening is assumed,
an iterative procedure is required to determine the plastic strain
at the current time step. Having determined Dcn, one can update the plastic strains and the hardening parameter, and the
stress at the current time step is calculated using updated
parameters.
The increment of plastic damage parameter, D-n+1, is updated
with the help of plastic increment computations. Total damage
parameter can be obtained as: -n + D-n+1 = -Zn+1. The new value
of the damage parameter is used to obtain the studied parameters
for the new increment of loading. In order to obtain the trial elastic
solution of the model, Eqs. (30) and (31) were modified to
Fig. 6. Ratcheting behavior of the thin cylinder for smax = 2.928 and rp = 0.5,
material models constants and thin cylinder geometry are given in Table 1. (a)
Variation of the maximum nondimensional tangential plastic strain, eph , at the outer
surface of the thin cylinder as a function of the number of thermal loading cycles
and (b) normalized equivalent stress versus normalized equivalent plastic strain at
the outer surface.
302
A. Nayebi / Materials and Design 31 (2010) 296–305
8
Bree’s biaxial results [7]
Coupled damage – NLKH results
7
6
τ max
5
4
Ratcheting Domain
3
Plastic Shakedown
2
Shakedown in in and out surfaces
Shakedown in out surface &
Elastic in inner surafce
1
Elastic Domain
0
0
0.1
0.2
0.3
0.4
0.5
0.6
σp
0.7
0.8
0.9
1
1.1
1.2
Fig. 7. Interaction diagram for thermal load parameter, smax, versus internal pressure loading parameter, rp, solid line (—) shows the two-dimensional Bree’s diagram and the
dashed line () presents the new results based on the continuum damage mechanics.
0
r þ 1
B p ð1t2 Þ
STrial;nþ1
¼ ð1 -n Þ@
h
hR 1
2
1
2
i
p;n
ð1 -n Þðep;n
h þ tez þ sn Þdg
R 12
1 ð1 -n Þdg
2
1
1
ð1 -n Þ
C
ðep;n þ tep;n
z Þ sn A þ R 1
ð1 t2 Þ h
2
ð1 - Þdg
1
2
Z
1
2
1
2
n
ð1 -n ÞDsdg ð1 -n ÞDs
0
B
¼ ð1 -n Þ@
STrial;nþ1
z
rp þ ð11t2 Þ
hR 1
2
1
2
ð36Þ
i
p;n
ð1 -n Þðep;n
z þ teh þ sn Þdg
1
R2
1 ð1 -n Þdg
2
1
1
ð1 -n Þ
C
ðep;n þ tep;n
h Þ sn A þ R 1
ð1 t2 Þ z
2
ð1 - Þdg
1
2
Z
1
2
1
2
ð1 -n ÞDsdg ð1 -n ÞDs
With the help of the incremental form of Eqs. (14), (15), (30),
(31), and (34) as below and using Newton–Raphson method for
Eq. (33), the plastic strains increment can be determined
p
enþ1 ¼ eTrial
nþ1 þ Denþ1
p
p
enþ1 ¼ epn þ Denþ1
ð38Þ
2
X nþ1 ¼ X n þ ð1 -nþ1 Þ C Depnþ1 cX nþ1 Dpnþ1
3
s
Y nþ1
-nþ1 ¼ -n þ
Dpnþ1
S
12
2
2 2 Dpnþ1 ¼ pffiffiffi Dezp;nþ1 þ Dehp;nþ1 þ Dezp;nþ1 Dehp;nþ1
3
~ nþ1
~ nþ1
Dep;nþ1
2r
r
2X z þ X h
z
h
z
¼
p;nþ1
nþ1
~
~ nþ1
2rh r
2X h þ X z
Deh
z
ð39Þ
ð40Þ
ð41Þ
ð42Þ
ð43Þ
n
ð37Þ
Ds is the increment of nondimensional temperature difference
across the thickness. It is assumed that the maximum pressure is
not greater than the yield pressure and elastic solution is only
needed for pressure loading. The loading associated with the thermal gradient is cyclic and can be considered with the constant pressure loading. With these trial stresses, the yield function (Eq. (33)) is
verified. If the yield criterion is violated, plastic solution is used.
Table 2
Material models constants and applied loadings.
I
II
III
IV
V
VI
c
Q (MPa)
q
smax
rp
Fig.
250
250
250
250
50
50
0.06
0.6
0.06
0.6
0.6
0.6
2
0.5
2
0.5
2
2
2.928
2.928
2.196
2.196
2.196
2.928
0.5
0.5
0.5
0.5
0.5
0.5
8a
8b
8c
8d
9a
9b
A. Nayebi / Materials and Design 31 (2010) 296–305
0
hR 1
0
hR 1
i
p;nþ1
1
2
r
þ tep;nþ1
þ snþ1 dg
p þ ð1t2 Þ
z
1 ð1 -nþ1 Þ eh
B
2
Snþ1
¼ ð1 -nþ1 Þ@
h
R 12
1 ð1 -nþ1 Þdg
2
1
1
C
snþ1 A
ð44Þ
ep;nþ1 þ tep;nþ1
z
ð1 t2 Þ h
r þ 1
B p ð1t2 Þ
Snþ1
¼ ð1 -nþ1 Þ@
z
2
1
2
i
ð1 -nþ1 Þ ep;nþ1
þ tep;nþ1
þ snþ1 dg
z
h
R 12
1 ð1 -nþ1 Þdg
2
1
1
C
snþ1 A
þ tep;nþ1
ep;nþ1
z
h
2
ð1 t Þ
ð45Þ
5. Results and discussion
The constitutive model parameters for 2 14 CrMo steel at 580 °C
were given by Lemaitre and Desmorat [16] and are shown in
Table 1. Model constants are temperature dependent but the average values were chosen and it was assumed that the mean temperature is constant during loading and unloading. The mean radius, R,
and thickness, t, are 100 mm and 10 mm, respectively.
Constant mechanical and cyclic thermal loading were applied.
Maximum inside pressure (mechanical loading) was not allowed
to exceed the yield pressure. Starting from zero, an incremental
thermal loading with a linearly varying temperature difference distribution across the wall thickness was applied. When the linear
temperature gradient attained its maximum, it was reduced incre-
303
mentally to zero. At this point, a full thermal stress cycle is completed. The number of loading increments varies in different load
cases in the test matrix, as each increment applies less than 1 °C
in temperature gradient. Up to 100 load increments per cycle
and up to 1000 cycles are applied in the most severe temperature
gradients. In some cases, a steady cyclic state stress–strain plot is
attained after the first cycle.
As an example, the amplitude of the nondimensional cyclic
thermal gradient across the thin wall of the cylinder, smax, and
the nondimensional constant mechanical stress, rp, were assumed to be 1.976 and 0.5, respectively. Fig. 4a shows the variation of the tangential plastic strain at the outer surface of the thin
cylinder as a function of the number of cycles. Plastic strain stays
constant during cyclic loading. Fig. 4b shows the variation of the
equivalent stress as a function of equivalent plastic strain at the
outer surface. The thin cylinder behavior is elastic after the first
cycle and elastic shakedown was obtained in the first cycle. The
amplitude of the nondimensional temperature gradient was increased to smax = 2.196 and nondimensional mechanical stress
was not changed (rp = 0.5). Plastic shakedown was obtained
(Fig. 5a and b). Fig. 5a shows that the nondimensional tangential
plastic strain at the outer surface stays constant after 62nd cycle
and the equivalent stress-equivalent plastic strain loop does not
evolve (Fig. 5b). Finally, as it was shown in Fig. 6a, when the cyclic and constant loadings are smax = 2.928 and rp = 0.5, respectively, the plastic strain at the outer surface increases with a
constant slope in each loading cycle. Fig. 6b shows that the
stress–strain loop evolves and ratcheting phenomenon was
resulted.
Fig. 8. Variation of the maximum nondimensional tangential plastic strain, eph at the outer surface as a function of the number of thermal loading cycles for smax = 2.928 and
rp = 0.5: (a) Q = 0.06 MPa (Table 2, case I), (b) q = 0.5 (Table 2, case II), and smax = 2.196 and rp = 0.5: (c) Q = 0.06 MPa (Table 2, case III) and (d) q = 0.5 (Table 2, case IV) (for
each case, other material constants are given in Table 1).
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A. Nayebi / Materials and Design 31 (2010) 296–305
Different results were obtained for the inner and outer surfaces
of the thin cylinder, with changing the applied mechanical and
thermal loading. In order to identify different regions in the Bree’s
diagram, more than 100 combinations of mechanical and thermal
loadings were simulated. The Bree’s diagram is divided into different regions as a function of the first yield situation and the inner
and outer surface of the thin cylinder behavior. These regions are
shown in Fig. 7 which is known as the Bree’s diagram. In order
to obtain the interaction diagram for Bree’s results [7], the inner
pressure and the temperature gradient were normalized according
to Eqs. (32).
It should be noted that the Bree’s model was obtained using the
Tresca’s criterion. In the present study, the von-Mises criterion was
used. In order to be able to compare the results, the inner pressure
in this
study was normalized using the following relation:
pffiffi
rp ¼ 23 tPrry .
Region A in Fig. 7 corresponds to the elastic answer. The outer
surface of the thin cylinder yields and elastic shakedown begins
with increasing the temperature difference (region B). The inner
surface remains elastic. In the third region C, the inner surface also
yields and both surfaces are in elastic shakedown. When maximum
thermal stress reaches 2ry, the plastic strain loops are formed and
plastic shakedown is obtained (region D). For the thermal and
mechanical stresses in the region D, the inner and outer surfaces
have the same behavior and both are in plastic shakedown state.
Finally, the ratcheting was obtained for two directions in the thin
walled cylinder, in the region E. Continuum damage mechanics,
which is dependent on the accumulated plastic strain, limits the
plastic shakedown zone. The new boundaries were obtained by
applying constant internal pressure while the temperature gradient was increased gradually to obtain each boundary between elastic, elastic shakedown, plastic shakedown, and ratcheting.
As it was shown in the Bree’s diagram (Fig. 7), the two dimensional model of Bree can not predict the ratcheting because of the
damage progress due to the accumulated plastic strain. But the 2ry
limit of the shakedown behavior is independent of the models and
the material behavior. Taking into consideration the nonlinear
kinematic hardening behavior into the model, leads to the shift
of the boundary between the plastic shakedown and ratcheting
for greater internal pressures.
It should be noted that other simulations have been conducted
using different values of the damage model constants, Q and q.
Although, the rate of ratcheting and the number of cycles before
shakedown were changed, the same boundaries between different
material behaviors were obtained. In the following simulations,
one of the parameters, Q, q and c were changed in each example,
and other material models parameters were as the same values
in Table 1. Table 2 gives the used material models constants in
these simulations and Figs. 8 and 9 show the variation of the nondimensional plastic strain as a function of the number of thermal
loading cycles. For the fist simulation, material damage model constant, Q, was decreased to 0.06 with respect to the simulations of
Fig. 7. The cyclic thermal and constant mechanical loadings are
smax = 2.928 and rp = 0.5, respectively (case I, Table 2). For the
same loading, ratcheting behavior had been resulted for Q = 0.6
(see Fig. 6) and the same behavior was obtained for Q = 0.06
(Fig. 8a). In order to show the effect of the other material damage
constant, q, on the thin cylinder behavior, the loadings were not
changed and q was decreased to 0.5 (case II, Table 2). For this case,
Fig. 8b shows that the behavior of the thin cylinder under theses
loadings was not changed.
Two other examples were considered for shakedown behavior.
The loadings were changed to smax = 2.196 and rp = 0.5, and Q
parameter was set to 0.06 (case III, Table 2). For the next simulation, the loadings were not changed and q parameter was diminished to 0.5 (case IV, Table 2). In both cases, plastic shakedown
behavior was obtained (Fig. 8c and d). It was shown in Fig. 5 that
the behavior was also plastic shakedown for the same loadings.
The effect of the, c, constant of the kinematic hardening model
was studied with changing it to 50. For two different above loadings (case V and VI, Table 2), the behavior of the thin cylinder
was unchanged. As it was shown in Fig. 9a, number of cycles
needed for plastic shakedown, was increased from 62 (Fig. 5a) to
300, and the rate of ratcheting was increased (Fig. 9b).
6. Conclusions
Fig. 9. Variation of the maximum nondimensional tangential plastic strain, eph at the
outer surface as a function of the thermal loading cycles when nonlinear kinematic
hardening constant, c, was decreased to 50 and the loadings are (a) smax = 2.196 and
rp = 0.5 (Table 2, case V), (b) smax = 2.928 and rp = 0.5 (Table 2, case VI) (for each
case, other material constants are given in Table 1).
The results of Kang et al. [15] showed that the coupling continuum damage mechanics and cyclic constitutive models, can improve the prediction of materials behavior in cyclic loading.
Motivated by their results, the behavior of the Bree’s cylinder
was studied with combining the unified continuum damage law
and the nonlinear kinematic hardening model. The behavior of thin
cylinder subjected to the constant internal pressure and the cyclic
temperature gradient was studied. The effect of damage was considerable and the plastic shakedown domain obtained for low pri-
A. Nayebi / Materials and Design 31 (2010) 296–305
mary stresses and cyclic temperature by Bree [7] was modified.
Taking into consideration the hardening effect, which was neglected in Bree’s analysis, shifts the plastic shakedown boundary
for higher internal pressure. The role of the material constants
was also studied. It was shown that the variation of these constants
did not change the behavior of materials, however, the rate of the
accumulated plastic strain and cycles needed to obtain plastic
shakedown were affected. It should be noted that because of the
creep strains, the continuum damage mechanics of the creep
mechanisms must also be considered in order to obtain more realistic boundaries in the interaction diagram. This will be the subject
of a future research.
Acknowledgments
Discussions and advice of Professor M. Mahzoon (Shiraz University) and Dr. M. Dadfarnia (University of Illinois at Urbana Champaign) are appreciated.
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